The Remainder Theorem

Lesson 2 of 7: Arithmetic with Polynomials

In this lesson:

• Discover that remainder = p(a) when dividing by (x − a)
• Use the Factor Theorem to find factors from zeros
• Use synthetic division to factor polynomials efficiently

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. State the Remainder Theorem: when is divided by , the remainder equals
2. Use the Remainder Theorem with synthetic division to evaluate polynomials efficiently
3. State and apply the Factor Theorem: is a factor if and only if
4. Test potential rational roots using the Factor Theorem
5. Explain why the Remainder Theorem is true from the division equation

Recall: Division and Polynomial Evaluation

• Division algorithm: (dividend = divisor × quotient + remainder)
• Remainder: the constant left over — not the quotient
• Evaluate : substitute into

Quick check: what is for ?

What Is the Remainder When You Divide?

Integer division: remainder

Check: ✓

Polynomial division: divide by — what is the remainder?

Can you find it without doing the full division?

Integer and Polynomial Division: Same Structure

Remainder Is Always a Constant Here

When is divided by (degree 1):

• Remainder degree : so is a constant
• Identity holds for every — pick

Long Division Confirms the Structure

Divide by :

Remainder = 7

Now evaluate directly: ✓

Quick Check: Long Division Practice

Divide by :

1. What is the quotient?
2. What is the remainder?
3. Verify: evaluate and confirm it equals your remainder.

Try it before moving on.

The Remainder Theorem: No Division Needed

Remainder Theorem: For any polynomial and any number :

The remainder when is divided by equals .

Shortcut: To find the remainder — just evaluate .

The Proof Walks Through One Step

The division equation is true for all :

Substitute — valid because the equation holds everywhere:

Therefore: ■

Remainder Theorem: First Worked Example

Find the remainder when is divided by .

By the Remainder Theorem: remainder

Remainder

(No long division needed.)

Remainder Theorem Example Two: Sign Alert

Find the remainder when is divided by .

, so

Remainder

Divisor means , not .

Remainder Theorem: Third Worked Example

Find the remainder when is divided by .

Remainder

Missing powers (like here) have coefficient 0.

Find the Sign Error in This Solution

A student evaluates the remainder when is divided by :

"Remainder = p(3) = 27 − 15 + 1 = 13"

What is wrong? What should the student have done?

Identify the error and give the correct remainder.

Practice: Apply the Remainder Theorem Now

Find the remainder when is divided by :

1. What is ?
2. Evaluate — that is the remainder.

No division needed.

Factor Theorem: Zeros Equal Factors

Factor Theorem (corollary of Remainder Theorem):

is a factor

• If: remainder exact division
• Only-if:

Test Both Directions of the Factor Theorem

Direction 1: — is a factor?

Direction 2: Is a factor of ?

— No.

Decide Direction 1 before advancing.

Factor Theorem: Factoring a Cubic

Factor :

✓ → is a factor

Divide → quotient

Factoring When Leading Coefficient Isn't One

Factor :

✓ → is a factor

Quotient:

Zeros:

Rational Root Theorem: Finite Candidates

Rational zero : ,

For :

From Factor Theorem to Synthetic Division

Factor Theorem: confirms a factor exists.

Synthetic division: computes the quotient.

• Divide out to reduce the degree
• Final number always equals — built-in check

Synthetic Division: The Layout and Steps

Divide by , :

• Write at left; coefficients across the top
• Bring down → multiply by → add → repeat

Last number = remainder =

Synthetic Division: Walk Through the Steps

Divide by :

Quotient: , Remainder:

Synthetic Division: When Remainder Is Zero

Remainder → is a factor of

Your Turn: Factor This Cubic Completely

1. Find a zero using the Factor Theorem.
2. Use synthetic division to get the quotient.
3. Factor the quotient.

No hints — do the full procedure yourself.

Two Zeros: Partial Factored Form

Suppose you test and find:

Without computing the full factorization: what do you know about the factored form of ?

Write the partial form. How many total factors does a cubic have?

Three Common Errors to Watch For

Wrong sign: , not

Bottom row: last number = remainder; rest = quotient

RRT isn't exhaustive: no rational roots ≠ no real roots

Evaluation and Division Are Connected

✓ Remainder Theorem: dividing by gives remainder

✓ Factor Theorem: is a factor

✓ Workflow: find zero → name factor → divide → factor quotient

is the zero of the divisor — not the divisor itself

Next: Zeros Reveal the Graph's Shape

You can now find all zeros of a polynomial using the Factor Theorem.

The next question: what does the graph do near each zero?

• Cross the -axis, or bounce off it?

That depends on how many times each factor appears.

APR.B.3 — Graphing Polynomials from Zeros